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Information Systems: Help and Review19 chapters | 387 lessons | 4 flashcard sets

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Lesson Transcript

Instructor:
*Michael Quist*

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Binary math uses a base 2 number system, and is the foundation of the on-off way of thinking for computers. In this lesson, we'll discover how to reach the decimal value 255 using binary numbers.

Briefly stated, you can get to **255** very quickly using **binary** (base 2) numbers, because 255 is 1 less than 256, which is 2 to the 8th power (2 x 2 x 2 x 2 x 2 x 2 x 2 x 2). Any power of 2 is expressed in binary as a 1 with a number of 0's following, equal to that power. For instance, 16 is 2^4 (or 1 with four zeroes in binary).

Power of 2 | Decimal Value | Binary Value |
---|---|---|

0 | 1 | 1 (no zeroes) |

1 | 2 | 10 (one zero) |

2 | 4 | 100 (two zeroes) |

3 | 8 | 1000 (three zeroes) |

4 | 16 | 10000 (four zeroes) |

5 | 32 | 100000 (five zeroes) |

6 | 64 | 1000000 (six zeroes) |

7 | 128 | 10000000 (seven zeroes) |

8 | 256 | 100000000 (eight zeroes) |

See the pattern? In binary, 2's are like 10's are in the decimal system. In the decimal system, 9 comes before 10. 99 comes before 100. 999 comes before 1000. In binary, it's all 1's and 0's. So, the step before 256 is all 1's. Therefore, 255 in binary is 11111111.

The step before any power of 2 is a string of 1's. Just like 9 comes before 10 in decimal, 1 comes before 10 in binary. There are as many 1's as the number of the power that is the next step.

Power of 2 | Decimal Value | Decimal Value Minus 1 | Binary Value |
---|---|---|---|

0 | 1 | 0 | 0 (no 1's) |

1 | 2 | 1 | 1 (one 1) |

2 | 4 | 3 | 11 (two 1's) |

3 | 8 | 7 | 111 (three 1's) |

4 | 16 | 15 | 1111 (four 1's) |

5 | 32 | 31 | 11111 (five 1's) |

6 | 64 | 63 | 111111 (six 1's) |

7 | 128 | 127 | 1111111 (seven 1's) |

8 | 256 | 255 | 11111111 (eight 1's) |

256 would be expressed in binary by 100000000, a 1 with eight 0's, since it's 2 to the eighth power, so binary 255 would be the step before that, which would be 11111111 (eight 1's in a row). Any number that is 1 less than a power of 2 will be merely a string of 1's equal to that number's power.

So again, we see that 255 in binary is written as 11111111.

The general steps for converting a decimal into a binary number are simple:

- Determine if the number is odd or even. If the number is odd. If the number is
**odd**, write a**1**for the next lowest digit and subtract 1 from the number. - If the number is
**even**, write a**0**for the next lowest digit and don't subtract anything. - Divide by 2.
- Continue these steps until you've reached 0.

Our 255 would come out like this:

- 255 is odd, so we start with a 1, subtract 1 from 255, and divide the 254 by 2.
- 127 is odd, so we write another 1, subtract 1 from 127, and divide 126 by 2.
- 63 is odd, so we write another 1, subtract 1 from 63, and divide 62 by 2.
- We would do the same steps for 31, 15, 7, 3, and 1.
- The final number would be 11111111.

Another way to express decimals in binary is to count through the binary numbers. Counting in binary is like counting in decimal, although you change orders of magnitude much more rapidly. It takes ten counts to get to ten, or '10' in decimal, but it only takes two counts to get to '10' in binary. It should be noted that the two values are not equivalent; 10 in binary is 2 in decimal. Here's a count to 16 in decimal, which is 10000 in binary:

Decimal | Binary |
---|---|

1 | 1 |

2 | 10 |

3 | 11 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

8 | 1000 |

9 | 1001 |

10 | 1010 |

11 | 1011 |

12 | 1100 |

13 | 1101 |

14 | 1110 |

15 | 1111 |

16 | 10000 |

When you're converting back to decimal, all you have to do is add up the values of each of the 1's in the binary number. For example, 11001100 has a 1 in the 4, 8, 64, and 128 positions. This means that 11001100 = 4 + 8 + 64 + 128 = 204 in decimal.

Binary numbers are numbers that are written in base two. This means that any power of two is written as a one with a number of zeroes following it and a whole number can be represented in binary. 256 is an even power of ten, equivalent to 100000000. Since it's one less than an even power of two, the number 255 can be written in binary as 11111111.

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